Use slopes of parallel and perpendicular lines

Ontario Grade 10 Mathematics

Analytic Geometry

Recognize how line direction is shown by slope, then use slope to compare lines.

Imagine two straight roads on a map that keep the same direction and never meet. They are like parallel lines. Two other lines might meet to form a square corner. They are perpendicular. Slope gives us a way to compare these line directions. First, we will review what slope tells us. Then we will use it to decide how lines are related.

What you will learn

Recall how slope describes a line’s steepness and direction.
Determine whether two lines are parallel by comparing their slopes.
Determine whether two non-vertical lines are perpendicular by using negative reciprocal slopes.
Recognize how horizontal and vertical lines fit these rules.

1. Grade 9 bridge: what slope tells us

Slope describes how a line changes as you move from left to right. It compares vertical change with horizontal change. The vertical change is called the rise. The horizontal change is called the run. A line that rises as you move right has a positive slope. A line that falls as you move right has a negative slope.

For two points on a line, subtract their

y

-coordinates to find the rise and their

x

-coordinates to find the run. Keep the point order consistent in both subtractions.

A horizontal line has no vertical change, so its slope is

0

. A vertical line has no horizontal change. Its slope is undefined because division by zero is not possible. Remember these cases when comparing lines.

m=\frac{y_2-y_1}{x_2-x_1}

Slope is rise divided by run.
A positive slope rises to the right; a negative slope falls to the right.
Horizontal lines have slope $0$ ; vertical lines have undefined slope.

2. Parallel lines: same direction, same slope

Parallel lines stay the same distance apart and do not meet. On a coordinate grid, they point in the same direction. Distinct, non-vertical parallel lines have equal slopes. For example, two lines with slope

2

are parallel if they are distinct.

You can read a line’s slope directly when its equation is in the form

y=mx+b

. In this form,

m

is the slope. The value of

b

tells where the line crosses the vertical axis; it does not change the slope.

Vertical lines are parallel to other vertical lines. Their slopes are both undefined, so compare their direction rather than treating undefined as a number. A vertical line is not parallel to a horizontal line.

m_1=m_2

Distinct non-vertical parallel lines have equal slopes.
Vertical lines are parallel to other vertical lines.
The vertical-axis crossing point does not change a line’s slope.

3. Perpendicular lines: slopes turn to form a square corner

Perpendicular lines meet at a right angle, which measures

90^\circ

. For two non-vertical lines, their slopes are negative reciprocals. A reciprocal is made by switching the numerator and denominator of a fraction. To make it a negative reciprocal, also change the sign.

For example, the negative reciprocal of

\frac{3}{4}

is

-\frac{4}{3}

. The negative reciprocal of

-2

, written as

-\frac{2}{1}

, is

\frac{1}{2}

. Both the sign and the fraction’s steepness change in the required way.

A horizontal line and a vertical line are also perpendicular. Their slopes are

0

and undefined, so use the directions of the lines rather than the negative-reciprocal calculation.

m_1m_2=-1

Perpendicular lines meet at a $90^\circ$ angle.
For non-vertical lines, switch the numerator and denominator and change the sign.
A horizontal line and a vertical line are perpendicular.

4. Guided example and independent practice

Use a steady routine when comparing lines. Find each slope from its equation or from two points. For parallel lines, check whether the slopes are equal. For perpendicular lines, check whether they are negative reciprocals. First look for a horizontal or vertical line, since those cases need special attention.

Try this on your own: a line has slope

-\frac{5}{2}

. What slope would a parallel line have? What slope would a perpendicular line have? Then decide whether a horizontal line and a vertical line are parallel, perpendicular, or neither. Use the rules to explain each answer.

Find the slopes before deciding how the lines are related.
Use equal slopes for parallel lines and negative reciprocal slopes for perpendicular lines.
Check horizontal and vertical lines separately.

Slope patterns for comparing lines

Line relationship	Slope pattern	Example
Parallel non-vertical lines	Equal slopes	$\frac{2}{3}$ and $\frac{2}{3}$
Perpendicular non-vertical lines	Negative reciprocal slopes	$\frac{2}{3}$ and $-\frac{3}{2}$
Parallel vertical lines	Both slopes are undefined	Vertical and vertical
Perpendicular horizontal and vertical lines	One slope is $0$ and the other is undefined	Horizontal and vertical

Worked example

Compare two lines from their equations

Line A is

y=\frac{3}{4}x+2

. Line B passes through

(1,5)

and

(5,2)

. Decide whether the lines are parallel, perpendicular, or neither.

Read Line A’s slope
Line A is written in the form $y=mx+b$ . In this form, the coefficient of $x$ is the slope, so Line A has slope $\frac{3}{4}$ .
$m_A=\frac{3}{4}$
Calculate Line B’s slope
Use the change in $y$ divided by the change in $x$ . Keep the two points in the same order in both parts of the fraction.
$m_B=\frac{2-5}{5-1}=-\frac{3}{4}$
Compare the slopes
The slopes are not equal, so the lines are not parallel. The negative reciprocal of $\frac{3}{4}$ is $-\frac{4}{3}$ , not $-\frac{3}{4}$ . Therefore, the lines are not perpendicular either.
$-\frac{3}{4}\ne\frac{3}{4},\qquad -\frac{3}{4}\ne-\frac{4}{3}$

Answer: The lines are neither parallel nor perpendicular.

Check: For a line perpendicular to one with slope

\frac{3}{4}

, the other slope must be

-\frac{4}{3}

. Line B’s slope is

-\frac{3}{4}

, so it does not meet that rule.

Common mistakes and how to avoid them

Calling two lines perpendicular just because their slopes have opposite signs.

Correction: Opposite signs are not enough. For non-vertical lines, the slopes must be negative reciprocals. For example,

\frac{3}{4}

and

-\frac{4}{3}

meet the rule.

Changing the sign but forgetting to switch the numerator and denominator.

Correction: For a negative reciprocal, do both actions. The negative reciprocal of

\frac{2}{5}

is

-\frac{5}{2}

.

Trying to calculate the slope of a vertical line by dividing by zero.

Correction: A vertical line’s slope is undefined. Compare vertical lines by their direction, and remember that a vertical line is perpendicular to a horizontal line.

Using different point orders for the rise and run.

Correction: Subtract coordinates in matching orders. Reversing the order in both numerator and denominator gives the same slope; reversing only one changes the sign incorrectly.

Lesson summary

Slope measures rise divided by run.
Parallel lines have equal slopes, with vertical lines handled as a separate direction case.
Perpendicular non-vertical lines have negative reciprocal slopes.
Horizontal and vertical lines are perpendicular; their slopes are $0$ and undefined.

Check your understanding

Question 1

A line has slope

\frac{4}{7}

. Which slope would a distinct parallel line have?

$-\frac{7}{4}$
$\frac{4}{7}$
$-\frac{4}{7}$
$\frac{7}{4}$

Show answer and explanation

\frac{4}{7}

Distinct non-vertical parallel lines have equal slopes, so the slope remains

\frac{4}{7}

.

Question 2

A line has slope

-\frac{3}{5}

. Which slope could belong to a perpendicular line?

$-\frac{5}{3}$
$\frac{3}{5}$
$\frac{5}{3}$
$-\frac{3}{5}$

Show answer and explanation

\frac{5}{3}

Switch the numerator and denominator, then change the sign. The negative reciprocal of

-\frac{3}{5}

is

\frac{5}{3}

.

Question 3

How are a horizontal line and a vertical line related?

They are parallel.
They are perpendicular.
They have equal slopes.
They are neither parallel nor perpendicular.

Show answer and explanation

They are perpendicular.

A horizontal line and a vertical line meet at a right angle, so they are perpendicular.

Key terms

Slope: A measure of a line’s direction and steepness, found by dividing vertical change by horizontal change.
Rise: The vertical change between two points.
Run: The horizontal change between two points.
Parallel lines: Lines that stay the same distance apart and do not meet.
Perpendicular lines: Lines that meet at a right angle of $90^\circ$ .
Reciprocal: A number formed by switching the numerator and denominator of a fraction.

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